Reconstruction of images from cone beam data

ABSTRACT

A computed tomography x-ray imaging system acquires a three-dimensional array of x-ray attenuation values using a cone beam x-ray source [(13)] and a curved two-dimensional array of detector elements [(16)]. Two-dimensional image slices [(55)] are reconstructed using a filtered back projection method, and corrections are made to the images to account for inaccuracies in the reconstruction method and for incomplete data due to the cone beam geometry.

BACKGROUND OF THE INVENTION

The present invention relates to computed tomography (CT) imagingapparatus; and more particularly, to reconstruction of images fromthree-dimensional data acquired with x-ray CT or SPECT scanners.

In a current computed tomography system, an x-ray source projects afan-shaped beam which is collimated to lie within an X-Y plane of aCartesian coordinate system, termed the "imaging plane." The x-ray beampasses through the object being imaged, such as a medical patient, andimpinges upon an array of radiation detectors. The intensity of thetransmitted radiation is dependent upon the attenuation of the x-raybeam by the object and each detector produces a separate electricalsignal that is a measurement of the beam attenuation. The attenuationmeasurements from all the detectors are acquired separately to producethe transmission profile.

The source and detector array in a conventional CT system are rotated ona gantry within the imaging plane and around the object so that theangle at which the x-ray beam intersects the object constantly changes.A group of x-ray attenuation measurements from the detector array at agiven angle is referred to as a "view" and a "scan" of the objectcomprises a set of views made at different angular orientations duringone revolution of the x-ray source and detector. In a 2D scan, data isprocessed to construct an image that corresponds to a two dimensionalslice taken through the object. The prevailing method for reconstructingan image from 2D data is referred to in the art as the filteredbackprojection technique. This process converts the attenuationmeasurements from a scan into integers called "CT numbers" or"Hounsfield units", which are used to control the brightness of acorresponding pixel on a cathode ray tube display.

In a 3D scan the x-ray beam diverges to form a cone beam that passesthrough the object and impinges on a two-dimensional array of detectorelements. Each view is thus a 2D array of x-ray attenuation measurementsand the complete scan produces a 3D array of attenuation measurements.Either of two methods are commonly used to reconstruct a set of imagesfrom the acquired 3D array of cone beam attenuation measurements. Thefirst method described by L. A. Feldkamp et al in "Practical Cone-BeamAlgorithm", J. Opt. Soc. Am., A/Vol. 1, No. 6/Jun. 1984 is a convolutionbackprojection method which operates directly on the line integrals ofthe actual attenuation measurements. The method can be implementedeasily and accurately with current hardware and it is a goodreconstruction for images at the center or "midplane", of the cone beam.The Feldkamp method employs the conventional convolution - backprojection form, but this is an approximation that becomes less accurateat larger cone beam angles. The second method proposed by PierreGrangeat in "Mathematical Framework of Cone Beam 3D Reconstruction Viathe First Derivative of the Radon Transform", Mathematical Methods InTomography, Herman, Louis, Natterer (eds.), Lecture notes inMathematics, No. 1497, pp. 66-97, Spring Verlag, 1991, provides anaccurate solution to the image reconstruction task based on afundamental relationship between the derivative of the cone beam planeintegral to the derivative of the parallel beam plane integral. Whilethis method is theoretically accurate, it requires mathematicaloperations that can only be solved using finite numerical calculationsthat are approximations. The errors introduced by the implementation ofthe Grangeat method can be greater than Feldkamp and these errors arenot correlated with cone beam angle.

SUMMARY OF THE INVENTION

The present invention relates to a computer tomography system whichproduces a three-dimensional array of data from which a set of 2D imageslices can be reconstructed. More specifically, the system includes a 2Darray of detector elements for receiving photons in a cone beam producedby a source while the two are rotated about a central axis to acquiredata at a series of views, an image reconstructor which employs filteredback projection of the acquired cone beam data to produce image dataf_(m0) (r); means for calculating a correction image f_(m1) (r) from theacquired cone beam data by filtering and backprojecting; and means forcombining the correction image data f_(m1) (r) with the back projectionimage data f_(m0) (r) to produce an image slice.

A general object of the invention is to accurately reconstruct imageslices from 3D cone beam data. A filtered back projection method isemployed to accurately and efficiently produce the main part of thereconstruction. A second set of image data is also produced by summingthe acquired cone beam data along each of its rows, filtering andweighting this summed data and back projecting it. The resultingcorrection image f_(m1) (r) is combined with the back projection imagef_(m0) (r) to produce the more accurate image slice.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a pictorial view of a CT imaging system in which the presentinvention may be employed;

FIG. 2 is a block schematic diagram of the CT imaging system;

FIGS. 3a and 3b are pictorial views of the cone beam produced by the CTimaging system;

FIG. 4 is a graphic illustration of the coordinate systems used todefine the reconstruction process; and

FIG. 5 is a block diagram of the image reconstructor which forms part ofthe CT imaging system of FIG. 2.

DESCRIPTION OF THE PREFERRED EMBODIMENT

With initial reference to FIGS. 1 and 2, a computed tomography (CT)imaging system 10 includes a gantry 12 representative of a "thirdgeneration" CT scanner. Gantry 12 has an x-ray source 13 that projects acone beam of x-rays 14 toward a detector array 16 on the opposite sideof the gantry. The detector array 16 is formed by a number of detectorelements 18 which together sense the projected x-rays that pass througha medical patient 15. Each detector element 18 produces an electricalsignal that represents the intensity of an impinging x-ray beam andhence the attenuation of the beam as it passes through the patient.During a scan to acquire x-ray projection data, the gantry 12 and thecomponents mounted thereon rotate about a center of rotation 19 locatedwithin the patient 15.

The rotation of the gantry and the operation of the x-ray source 13 aregoverned by a control mechanism of the CT system. The control mechanism20 includes an x-ray controller 22 that provides power and timingsignals to the x-ray source 13 and a gantry motor controller 23 thatcontrols the rotational speed and position of the gantry 12. A dataacquisition system (DAS) 24 in the control mechanism 20 samples analogdata from detector elements 18 and converts the data to digital signalsfor subsequent processing. An image reconstructor 25, receives sampledand digitized x-ray data from the DAS 24 and performs high speed imagereconstruction according to the method of the present invention. Thereconstructed image is applied as an input to a computer 26 which storesthe image in a mass storage device 29.

The computer 26 also receives commands and scanning parameters from anoperator via console 30 that has a keyboard. An associated cathode raytube display 32 allows the operator to observe the reconstructed imageand other data from the computer 26. The operator supplied commands andparameters are used by the computer 26 to provide control signals andinformation to the DAS 24, the x-ray controller 22 and the gantry motorcontroller 23. In addition, computer 26 operates a table motorcontroller 34 which controls a motorized table 36 to position thepatient 15 in the gantry 12.

As shown best in FIG. 3a, in the preferred embodiment of the presentinvention the detector array 16 is a flat array of detector elements 18,having N_(r) (e.g. 1000) elements 18 disposed along the in-plane (x,y)direction, and N_(z) (e.g. 16) elements 18 disposed along the z axis.The x-ray beam emanates from the x-ray source 13 and fans out as itpasses through the patient 15 and intercepts the detection array 16.Each acquired view is a N_(r) by N_(z) array of attenuation measurementsas seen when the gantry is oriented in one of its positions during thescan. As shown in FIG. 3B, the object of the present invention is toreconstruct as set of 2D image slices 35 from the 3D array of acquireddata produced by the x-ray cone beam during the scan. It can be seenthat because the cone beam diverges as it passes through the patient 15,the reconstruction of the parallel image slices 35 is not possible witha straight forward fan beam filtering and backprojection process. Thepresent invention enables a more accurate reconstruction of the imageslices 35 from this acquired cone beam data.

Cone beam tomographic imaging uses a 2D array detector to record 2Dline-projection measurements of a function to be reconstructed. Let thisfunction be denoted as f(r), where r is the position vector. Assume thatthe function f(r) has a finite support, denoted as Ω. Let Γ denote thescanning orbit. For simplicity, assume that the 2D detecting surfaceforms a flat plane that contains the origin O. Actual physical detectorarrangements can be converted to this form by mapping. A set of 2Dprojection measurements can be characterized by P_(os) (Y,Z), where thevertex position of a cone beam is described by OS and each ray withinthe cone beam is characterized by its intersection with the detectorplane, denoted as (Y,Z). Let X', Y' and Z' denote the unit vectors alongSO, the Y axis and the Z axis, where the Z axis is along the axis ofrotation.

Circular scanning, in theory, can not collect a complete set of data tosupport an exact reconstruction. In this case, the image function to bereconstructed, denoted as f(r), can be split into two terms:

    F(r)=f.sub.M (r)+f.sub.N (r)                               (1)

where f_(M) (r) and f_(N) (r) represent, respectively, the correspondingportion of f(r) that is supported and is not supported by the projectiondata. In most practical applications, especially when cone angle is notlarge, f_(N) (r) is small and a crude estimation of f_(N) (r) would besufficient. On the other hand, the image quality is primarily determinedby the quality of the f_(M) (r) reconstruction. It is highly desirable,therefore, to have a methodology for f_(M) (r) reconstruction that ismathematically rigorous and that can also be implemented accurately andcomputationally efficiently on commercially available systems.

It is a discovery of the present invention that f_(M) (r) in equation 1can be further split into two terms:

    f.sub.m =f.sub.M0 (r)+f.sub.M1 (r)                         (2)

where f_(M0) (r) represents an image reconstructed using the Feldkampmethod; and f_(M1) (r) represents the part of f_(M) (r) that issupported by the projection data but that is lost in the Feldkampreconstruction method. It is further discovered and set forth in detailin Appendix A that the formula for the f_(M1) (r) reconstruction can beexpressed as follows: ##EQU1## The Fσ_(os) (ω_(z)) is the Fouriertransform of σ_(os) (Z).

This reconstruction method is implemented in the image reconstructor 25.Referring particularly to FIG. 5, the cone beam projection data isreceived from the DAS 24 as a two-dimensional array of values which arepreprocessed in the standard manner at process block 40. Suchpreprocessing includes correcting for known errors and offsets andcalculating the minus log of the data to convert it to x-ray attenuationvalues.

The preprocessed cone beam attenuation profiles are used to separatelycalculate f_(M0) (r), f_(M1) (r) and f_(N) (r). The main image termf_(M0) (r) is calculated in a sequence of steps indicated by processblocks 41-44 which is essentially the method described by Feldkamp etal. It includes multiplying the cone beam projection data by weightingfactors, as indicated at process block 41: ##EQU2##

where d=distance from x-ray source to detector element.

The resulting projection data is then filtered by convolving it with afilter kernal as indicated at process block 42. ##EQU3##

where the kernals are: ##EQU4## The filtered attenuation data is thenback projected from each detector element position back along the rayextending from the point source of the x-ray cone beam. This results ina 3D image array f_(M0) (r). ##EQU5## where

    Y(r)=r. Y'd/(d+r. X')

    Z(r)=r. Z d/(d+r. X')

As is well known in the art, the image reconstructed in this mannerthrough the midplane of the cone beam is very accurate. However, as theimages move away from this midplane image, their quality decreases dueto errors in the methodology and incomplete data. Corrections for thisdeterioration in image quality is provided by the f_(M1) (r) and f_(N)(r) image terms described above and calculated as will now be described.

Referring still to FIG. 5, the correction image f_(M1) (r) is derivedfrom the weighted cone beam projections produced at process block 41 andindicated above in equation (7). This two-dimensional weighted cone beamprojection P_(OS) (Y,Z) is then summed along the row direction (Y) atprocess block 50 to produce the one-dimensional row sum σ_(OS) (Z)indicated above in equation (6). This is filtered at process block 51 bya one-dimensional filter (jω_(z)), as indicated above in equation (5) toproduce the filtered row sum p(z). This filtering can also be performedby directly differentiating the row sum σ_(OS) (Z), also as shown inequation (5). The filtered row sum for every projection in the scan isthen weighted by a position-dependent factor and the resulting data isback projected at process block 52 in accordance with the above equation(3) to produce the correction image f_(M1) (r).

The third term f_(N) (r) may be estimated using a number of knownmethods, but a methodology disclosed in the above-cited Grangeatpublication is presently preferred and is incorporated herein byreference. More specifically, Radon set S can be divided into twosubsets, subset D which is supported by projection data, and subset Cthat is not supported by projection data. Next, ∂Rf/∂ρ on the boundaryof subset D is calculated from the cone beam projection data using thefollowing equation disclosed by Grangeat that relates the derivative ofcone beam plane integral to the derivative of parallel beam planeintegral: ##EQU6## where, the line denoted by (1,θ) is the intersectionof the plane P characterized by (ρ=OS . n, n) and the detector planedenoted by OS. Next ∂Rf/∂ρ is estimated in subset C. Note that sincesubset C is not supported by cone beam projection data, we assume that∂Rf/∂ρ is continuous at the boundary of subset D and C, and interpolatetherebetween. Having calculated these values, f_(N) (r) may becalculated as follows: ##EQU7##

Referring to FIG. 5, to produce the corrective image f_(N) (r) theacquired and preprocessed cone beam attenuation data is applied to aprocess block 46 in calculated as set forth above in equation (11). Asindicated at process block 47, the data values in subset C are thencalculated by interpolating between values at the boundaries with subsetD, and these estimated values are applied to process block 48 whichcalculates the correction images f_(N) (r) using the above equation(12). The corresponding slices in each image f_(M0) (r), f_(M1) (r) andf_(N) (r) are added together at summing point 53 to produce the finalimage slices 55 for the computer 26.

APPENDIX A

In the following discussion reference is made to the coordinate systemand system geometry shown in FIG. 4. The circular scanning orbit, thoughincapable of collecting a complete set of data, is of great practicaland theoretical interests due to the simplicity and the symmetry itoffers. Consider a circular scanning orbit of radius d. In this case, OScan be described as:

    OS=(d cos φ, sin φ, 0)                             (A1)

Note that now the detector coordinates (1, Θ, OS) can also becharacterized as (1, θ, φ).

For a circular scan, set S(Γ) forms a torus described by the relation:ρ<d sin θ, and the image f_(M) (r) terms can be expressed in (ρ, θ, φ)as: ##EQU8## What follows is a proof that f_(M0) (r) is equivalent tothe Feldkamp image reconstruction. In set S(Γ), (ρ, θ, φ) and (1, θ, Φ)have the following relations: ##EQU9## Therefore, the Jacobean for thetransformation from (ρ, θ, φ) and (1, θ, φ) is ##EQU10## One then has:##EQU11## Furthermore, the derivative of the delta function in Equation(A2) can be rewritten as follows: ##EQU12## Using Equations (A7), (A8)and (A9), Equation (A2) can be rewritten in (1, θ, Φ) coordinate systemas: ##EQU13## Using the property of delta function, Equation (A10) canfurther be reduced to: ##EQU14## By examining equation (A11) one canconclude that f_(M0) (r) is equivalent to the Feldkamp method. Inaddition, using the property of delta function, equation (A3) can besimplified as follows: ##EQU15## Transforming from the sphericalcoordinate system (ρ, θ, φ) to the detector coordinate system (1, Θ, Φ),one has: ##EQU16## The Fσ_(os) (ω_(z)) is the Fourier transform ofσ_(os) (Z).

I claim:
 1. A computed tomography imaging system which comprises: atwo-dimensional array of detector elements for receiving photonsemanating in a cone beam from a source;a digital acquisition system foracquiring two-dimensional arrays of cone beam data from the array ofdetector elements at a series of views in which the array of detectorelements revolves around a central axis; a filter for receiving the conebeam data and filtering the same; means for back projecting the filteredcone beam data to produce image data f_(M0) (r); row summing means forreceiving the cone beam data and summing the values therein along one oftwo dimensions of said two-dimensional arrays of cone beam data to formone-dimensional arrays of row sum data; means for filtering the row sumdata; means for back projecting the filtered row sum data to producecorrective image data f_(M1) (r); and summing means for combining theimage data f_(M0) (r) with the correction image data f_(M1) (r) toproduce an image slice.
 2. The computed tomography system as recited inclaim 1 which includes:means for receiving the cone beam data andestimating values not provided by the received cone beam data; means forreceiving the estimated values and calculating corrected image dataf_(N) (r); and the summing means also combines the corrected image dataf_(N) (r) to produce the image slice.
 3. The system as recited in claim1 which includes means for weighting the cone beam data prior to itsapplication to the filter and the row summing means.
 4. The system asrecited in claim 1 in which the source produces x-rays and is located onthe side opposite the array of detector elements from the central axis,and both the array of detector elements and x-ray source are revolvedaround the central axis during the acquisition of said series of views.5. The system as recited in claim 4 in which the acquired cone beam datais preprocessed to provide x-ray attenuation data for athree-dimensional region about the central axis and between the x-raysource and the array of detector elements, and the image data f_(M0) (r)and corrective image data f_(M1) (r) is summed to produce a plurality oftwo-dimensional image slices through said three-dimensional region.